Wiener kernel

The Wiener kernel defines a non-stationary gaussian process, the Wiener process:

\begin{equation*} \displaystyle K(x, x') = \min(x, x') \end{equation*}

It is convenient to model time-series. It is the limit as \(n \rightarrow \infty\) of a random walk of length \(n\) (cite:rasmussen2003 p213). An interesting variant is the brownian bridge obtained by conditioning \(X(t_0)=0\) (cite:grimmett2020 p534).

The parameter \(l\) is the characteristic lengthscale of the process. As one can see on the figure below, the larger the value of \(l\) the "further" the kernel takes non-negligible values.